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Bit error rate performance analysis of ACMAP in multiple input single output wireless relay network
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 12 (2020)
Abstract
In this paper, we propose a joint decoding scheme called ACMAP decoder for multiple input single output (MISO) wireless cooperative communication network that consists of single source, single relay, and single destination. The proposed scheme is based on both Alamouti combining (AC) scheme and maximum a posteriori (MAP) decoder and is used to estimate the data at the destination. The ACMAP decoder is optimal in the sense that it minimizes the endtoend bit error rate (BER). In order to analyze performance of the proposed decoder, we derive a closed form expression for the upper bound (UB) on the endtoend error probability. Distances between system nodes, transmit energy, and channel noise and fading effects are considered in the derivation of the UB. Numerical results show that the closed form UB is very tight and it almost coincides with the exact BER results obtained from simulations. Therefore, we use the derived UB expression to study the effects of the relay position on the BER performance and to find the optimal location of the relay node.
Introduction
In recent years, cooperative communication is gaining a significant attention where relay nodes can collaborate with the users to enhance the wireless network performance. Cooperative relaying exploits the broadcast nature offered by the wireless medium where transmitted signals can be received, processed, and retransmitted by any node in the neighborhood of the source. The relay node could be a fixed node utilized by the network or another user that acts as a partner. In the second case, many partners may be available for each user to choose from which makes partner assignment important for better BER performance. Also, the energy allocation, for both user and relay in both cases, is important if the total energy is constrained. Cooperative communications provide a substantial improvement in the performance of the wireless networks in terms of rate (spectral efficiency or bandwidth) and reliability (diversity gain) [1–8]. This improvement can lead to the extension of the coverage and reduction in consumed energy. Cooperative relaying can have a great value in many systems such as ad hoc networks and next generation cellular and wireless local area networks. In cooperative diversity, relay nodes retransmit the signal received from the source which allows the receiver node to average channel variations resulting from fading and shadowing [1]. Several cooperation protocols have been proposed in the literature such as decode and forward (DF), amplify and forward (AF), and compress and forward (CF).
In this paper, we adopt the decode and forward (DF) cooperation protocol. In DF, the data transmission occurs over two phases. In the first phase, the source transmits its data to the intended destination. Because of the broadcast nature of the wireless medium, the signals from the source can be received by relay nodes. In the second phase, relay nodes decode the received signals and then forward the decoded data to the destination. Since the channel between the source and the relay is not necessarily errorfree, a decoding error may occur at the relay node. Therefore, the data received at the destination from the relay node may not provide the expected information about the data of the source. Accordingly, the diversity gain expected from the overall network may not be achieved. Several techniques were proposed to enable the destination to estimate the data transmitted by the source. In those techniques, the destination first combines the received signals and then uses the combined signal to detect the data of the source. Several combining techniques were proposed such equal gain combining (EGC), maximum ratio combining (MRC), and selection combining (SC). These combining techniques would provide a BER performance similar to that provided by traditional diversity techniques (i.e., when the destination receives multiple copies of the same signal directly from the source) if the link between the source and relay (SR link) is errorfree; otherwise, it will lead to an error floor [9]. Errorfree link can be achieved using automatic repeat request (ARQ) protocol at the relay node [10]. However, this will increase the overhead and, accordingly, reduces the network throughput. Several schemes have been proposed in the literature to decode the data at the destination taking into consideration the error in the SR link. The error performance of the cooperative communications with decodeandforward protocol with different combining techniques was also investigated in the literature. The endtoend performance of wireless communication systems with relays over Rayleigh fading channels was studied in [11] when the direct link between the source and the destination does not exist. A general framework for ML detection of both coherent and noncoherent uncoded cooperative diversity was presented in [12] where the authors derived a high SNR approximations based on the closedform BER expressions. An exact error analysis for decode and forward cooperation with maximal ratio combining in Nakagami fading was provided in [13]. The authors of [14] derived an analytical expression for the symbol error probability of the DF protocol with selection combining for Mary phaseshift keying in Rayleigh fading environment.
The maximum a posteriori (MAP) decoder was proposed and its BER performance was analyzed in [15, 16] for single input single output (SISO) wireless relay network. Unlike other decoders, the receiver does not use any combining techniques. However, it considers the data received from the source together with the data received from the relay node as a codeword. The decoding rule is to find the codeword that maximizes the a posterior probability. Since the error probability in the sourcetorelay link is not necessarily 0.5, the codewords received at the destination are not equiprobable. Therefore, the MAP decoding rule can not be simplified to the maximum likelihood (ML) decoding rule. Hence, the error probability of the SR link is considered in the decoding process and, accordingly, the MAP decoder provides optimal performance in terms of BER. In this paper, we first modify the MAP decoder to support MISO wireless relay network. We find that the straightforward modification of the MAP technique will increase the decoding complexity at the destination. Therefore, we propose a joint decoding scheme called ACMAP decoder that is based on both Alamouti combining (AC) scheme and maximum a posterior (MAP) decoder to estimate the data at the destination. The proposed scheme mitigates the complexity problem of the MAP decoder. The proposed scheme is optimal in the sense that it minimizes the endtoend BER. We also derive a closed from expression for the upper bound on the bit error probability of the decodeandforward cooperation protocol with the proposed ACMAP decoder. The derived upper bound takes into account the SNR of all links (i.e., sourcetorelay, sourcetodestination, and relaytodestination).
One of the end goals of this paper is to find the optimal position of the relay node. The problem of relay positioning and partner assignment was proposed in wireless cooperative and sensor networks to improve the overall system performance and energy efficiency. Optimal positioning of relay node is a very challenging and complex problem [17]. In order to address the complexity problem, two approaches were proposed in the literature. The first approach is to provide suboptimal solutions supported by heuristics [18]. The second approach is to find the optimal position by considering specific performance metrics [19, 20]. In this paper, we adopt the second approach to find the optimal position of the relay node in the case of fixed relay and to find the best partner if the source has many partners to choose from. The performance metric we use in this analysis is the bit error rate. Since the proposed ACMAP decoding scheme is optimal and the derived upper bound is very tight, the closed form expression for the UB on the BER is used to find the optimal location the relay node.
The remainder part of this paper is organized as follows. The system model is described in Section 2. The proposed decoding scheme is described in Section 3. The BER performance analysis for the ACMAP decoder is presented in Section 4. Positioning of relay node is presented in Section 5. Section 6 presents numerical results and discussions. Finally, the conclusions are drawn in Section 7.
System model
We consider a relay network composed of a source (S) equipped with two antennas, a relay (R), and a destination (D) as shown in Fig. 1. The data transmission occurs over two phases. In the first phase, the source sends its data to the destination using the Alamouti code [21] to achieve transmit diversity. Using this scheme, two symbols s_{0} and s_{1} are transmitted simultaneously twice in two time slots t_{0} and t_{1}. In time slot t_{0}, the two antennas of the source A_{1} and A_{2} transmit signals corresponding to s_{0} and s_{1}, respectively. In the next time slot t_{1}, the two antennas A_{1} and A_{2} transmit signals corresponding to −s_{1}^{∗} and s_{0}^{∗}, respectively, where “ ^{∗}” denotes the complex conjugate. Due to the broadcast nature of the wireless medium, the relay also receives the data from the source (possibly with some errors). We assume that all data are sent using BPSK modulation scheme and the source generates its bits with equal probability, i.e., p(bs=0)=p(bs=1)=0.5 where b_{s} is the source bit. We assume that all channels are Rayleigh flat fading with additive white Gaussian noise (AWGN). The channel gain is assumed to be constant over two consecutive time slots which is necessary for the decoding of the Alamoutitransmitted signals. The signals received at the end of the first phase (two time slots) by the relay and the destination, respectively, are
where
y_{sr0} is the signal received at the relay node in the first phase in time slot t_{0}.
y_{sr1} is the signal received at the relay node in the first phase in time slot t_{1}.
y_{sd0} is the signal received at the destination in the first phase in time slot t_{0}.
y_{sd1} is the signal received at the destination in the first phase in time slot t_{1}.
\(s_{i} \in \{+\sqrt {E_{s}},\sqrt {E_{s}}\}\) is the BPSK modulated signal of the ith symbol sent by the source, i∈{0,1}, where E_{s} is the transmit energy per the source bit.
h_{srj} and h_{sdj},j∈{0,1}, are the channel fading gains of the SR and SD links, respectively.
d_{sr} and d_{sd} are the lengths of the SR and SD links, respectively.
m is the path loss exponent.
n_{ri},i∈{0,1} is the AWGN noise at the relay node which has zero mean and variance N_{0}/2.
n_{sdi} is the AWGN noise at the destination node which has zero mean and variance N_{0}/2.
In the second phase, the relay node decodes the data received from the source using Alamouti combining scheme by first calculating the decision variables r_{0} and r_{1} as follows
and then estimates the transmitted bits as follows
where br_{i} represents an estimate of the source bit bs_{i} at the relay node. Then, the relay forwards the decoded data to the destination in two successive time slots. The signals received at the destination from the relay are
where
y_{rd0} is the signal received at destination from the relay node in the first time slot of the second phase (third time slot in total).
y_{rd1} is the signal received at destination from the relay node in the second time slot of the second phase (fourth time slot in total).
\(\hat {s}_{i} \in \{+\sqrt {E_{r}},\sqrt {E_{r}}\}\) is the BPSK modulated signal of the ith symbol estimated and sent by the relay, i∈{0,1}, where E_{r} is the transmit energy per the relay bit.
h_{rdi} is the channel gain of the RD in the ith time slot of the second phase, where i∈{0,1}.
d_{rd} is the length of the RD link.
n_{rdi} is the AWGN noise at the destination node which has zero mean and variance N_{0}/2.
Let the random vector e=[ e_{0} e_{1}]^{T} where e_{i}∈{0,1} captures the error events on the sourcetorelay channels, i.e., \(b_{r_{i}} = b_{s_{i}} \oplus e_{i}\), where ⊕ denotes the binary XOR (exclusive OR) operation. Hence, e_{i}=1 means br_{i}≠bs_{i}, i.e., the source bit is received in error at the relay node. In the case of BPSK modulation over flat fading channel with AWGN, the probability of error per bit in the sourcerelay link, i.e., P(e_{i}=1), is given by [22]
where \(\gamma _{sr}=d_{sr}^{m} E_{s}/N_{0}\) is the average receive SNR of the link between the source and the relay.
The received signals at the destination can be written in the matrix form as follows
where the received vector \(\mathbf {y}=[\!y_{sd0} \ y_{sd1}^{*} \ y_{rd0} \ y_{rd1}]^{T}\), the transmitted vector \(\mathbf {s}=[\!s_{0} \ s_{1} \ \hat {s}_{0} \ \hat {s}_{1}]^{T}\), the noise vector \(\mathbf {n}=[\!n_{sd0} \ n_{sd1}^{*} \ n_{rd0} \ n_{rd1}]^{T}\), and the channel matrix H is given by
Decoding schemes used at the destination
In the considered MISO cooperative communication network, the source node is equipped with two antennas while the relay and the destination nodes are equipped with single antenna. To achieve diversity gain, the source transmits its data using Alamouti space time block code (STBC). The relay node uses the decode and forward (DF) cooperation protocol in order to increase the reliability of the source data at the destination. Hence, the destination receives four signals (two from the source and two form the relay node) in four time slots. The destination uses these four signals jointly to decode the data sent from the source (i.e., \(b_{s_{0}}\) and \(b_{s_{1}}\)). In this section, firstly, we present the maximum a posterior (MAP) decoding scheme for MISO relay network. Secondly, we propose a new decoding scheme that mitigates the complexity problem of the MAP decoder.
Maximum a posterior decoder
In this section, we present the MAP decoding scheme used by the destination to estimate the data sent from the source for the MISO relay network under consideration. The MAP decoding scheme is optimal in the sense that it minimizes the error probability at the destination. Let b=[ bs_{0} bs_{1} br_{0} br_{1}]^{T} represents the bits vector (codeword) of the data transmitted from the source and relay nodes. If the SR link is errorfree, then br_{0}=bs_{0} and br_{1}=bs_{1} and, accordingly, there will be only four possible values of this vector. Since the SR link is not errorfree, there are sixteen possible values for this vector ranging from b_{1}=[ 0 0 0 0]^{T} to b_{16}=[ 1 1 1 1]^{T} based on the values of \(b_{s_{i}}\) and e_{i}. Since, in general, P(e_{i}=1)≠P(e_{i}=0), these 16 vectors (codewords) are not equiprobable. For example, the probability of transmitting the vector b_{5}=[ 0 1 0 0]^{T} is given by
Since the source bits are independent and equiprobable, then
The first part of (13) is given by
Substituting from (15) and (14) into (13) yields
Similarly, the probability of transmitting a specific vector (codeword) b_{i} is given by
The MAP decoder finds the transmitted vector b_{i} that maximizes the a posterior probability P(b_{i}y). An estimate of the transmitted vector is given by
and the maximization process is performed over the 16 possible vectors (codewords). Applying Bayes theorem to (18) yields
where [23]
and s_{i} is the modulated vector that corresponds to the vector b_{i}, e.g., \(\mathbf {s}_{3}=[\sqrt {E_{s}} \ \sqrt {E_{s}} \ \sqrt {E_{r}} \ \sqrt {E_{r}}]^{T}\). Substituting from (20) into (19) yields
For the described system model, in order for the destination to decode the two bits sent from the source, it has to search a space composed of sixteen vectors each of length four bits. Although the MAP decoder is optimal in the sense of minimizing the error rate, its complexity is high especially when the number antennas, network nodes, and/or the modulation order increases. In the following section, we present a less complex joint decoding scheme that combines both MAP and Alamouti decoding scheme at the destination to retrieve the source bits.
Proposed ACMAP decoder
In this section, we present the joint decoding scheme (ACMAP decoder) that is based on both Alamouti combining scheme and MAP decoder. The ACMAP decoder mitigates the complexity problem through utilization of the Alamouti combining scheme in reducing the size of the search space used by the MAP decoder while maintaining the same optimal BER performance. The ACMAP scheme is a twostage decoder. In the first stage, Alamouti combining scheme is used on y_{sd0} and y_{sd1} given by (3) and (4) to get two signals corresponding to s_{0} and s_{1} as follows
In the second stage, MAP decoder is used to retrieve the source symbols separately where d_{0} and y_{rd0} are used to decode s_{0} and d_{1} and y_{rd1} are used to decode s_{1}. The signals used for decoding s_{0} can be written in the matrix form as follows
where \(\mathbf {y}=[\!d_{0} \ y_{rd0}]^{T}, \mathbf {s}=[\!s_{0} \ \hat {s}_{0}]^{T}\), equivalent noise vector \(\mathbf {w}=[\!h_{sd0}^{*}n_{sd0}+h_{sd1}n_{sd1}^{*} \ n_{rd0}]^{T}\), and the equivalent channel matrix H is given by
Let b=[ bs_{0} br_{0}]^{T} represents the transmitted bits vector corresponding to s_{0}. Since the SR link is not errorfree, there are four possible values for this vector b_{1}=[ 0 0]^{T}, b_{2}=[ 0 1]^{T}, b_{3}=[ 1 0]^{T}, and b_{4}=[ 1 1]^{T} based on the values b_{s0} and e_{0}. Since, in general, P(e_{0}=1)≠P(e_{0}=0), these four vectors are not equiprobable. The probability of transmitting a specific vector b_{i} is given by
The MAP decoder finds the transmitted vector b_{i} that maximizes the a posterior probability P(b_{i}y). An estimate of the transmitted vector is given by
Applying Bayes theorem on (27) yields
where P(yb_{i}) has a multivariate normal distribution given by
the 2×2 covariance matrix Σ is given by
and s_{i} is the modulated vector corresponding to the vector b_{i}, e.g., \(\mathbf {x}_{3}=[\sqrt {E_{s}} \ \sqrt {E_{r}}]^{T}\). In order to simplify the calculations of (28) and the analysis of the bit error rate, we redefine the received vector y as follows
Hence, the covariance matrix of the noise vector would be Σ=(N_{0}/2)ι_{2}, where ι_{2} is the 2×2 identity matrix, and the equivalent channel matrix H would be given by
Therefore, (28) can be rewritten as follows
Similarly, the signals d_{1} and y_{rd1} can be used to decode s_{1}. Using two signals to decode each symbol reduces the size of the search space to four vectors each of length of two bits and, accordingly, reduces the decoding complexity. Since the first stage (Alamouti combining) preserves all the information about the data transmitted by the source, both MAP and ACMAP decoding schemes would provide the same performance. Figure 2 shows simulation results for the BER against the source transmit SNR (E_{s}/N_{0}) when the relay transmit SNR (E_{r}/N_{0}) is constant for both MAP and ACMAP decoders. As expected, the figure shows that the both MAP and ACMAP decoding schemes provide exactly the same BER performance.
The proposed ACMAP decoding scheme can be easily modified to support higher order modulation. However, the complexity will dramatically increase because of the massive increase in the size of the search space. For example, in 16QAM, the search space is composed of 256 vectors (codewords) instead of 4 in the case of BPSK.
BER analysis of the ACMAP decoder
In this section, we derive the upper bound (UB) on the bit error probability of the ACMAP decoding scheme.
Derivation of the upper bound
The destination uses (33) to estimate the transmitted codeword. Since the first bit of the codeword is an estimate of the bit coming directly from the source, the bit error probability would be given by
where P(b_{k}→b_{l}) is the pairwise error probability of confusing b_{k} with b_{l} when b_{k} is transmitted and when these are the only two hypothesis. When the vector b_{k} is transmitted, the received vector would be
and, according to (33), the probability P(b_{k}→b_{l}) would be given by
When h=[ h_{sd0} h_{sd1} h_{rd0}]^{T} is given, <w,H(s_{k}−s_{l})> would be a Gaussian random variable with zero mean and variance \(\frac {N_{0}}{2}H(\mathbf {s}_{k}\mathbf {s}_{l})^{2}\). Accordingly,
where α_{kl}=P(b_{k})/P(b_{l}). From, (26), there will three possible values for α_{kl} as follows
From (37cl_given+hTacx >), we notice that the probability P(b_{k}→b_{l}h) depends on: 1 The Hamming distance between b_{k} and b_{l},w_{kl}, (i.e., the number of positions at which b_{k} and b_{l} are different). 2 Whether b_{k} and b_{l} have the same probability.
The four possible codewords are b_{1}=[ 0 0]^{T}, b_{2}=[ 0 1]^{T}, b_{3}=[ 1 0]^{T}, and b_{4}=[ 1 1]^{T}. From (26), we find that codewords with equal probability (i.e., α_{kl}=1) have a hamming distance w_{kl}=2 (e.g., codewords b_{1} and b_{4}). Accordingly, and from (37cl_given+hTacx >) and (38), it is straightforward to show that P(b_{k}→b_{l}) does not depend on α_{kl} when w_{kl}=2. Let P(b_{k}→b_{l})=P_{1}(α_{kl}) when the hamming distance w_{kl}=1 and P(b_{k}→b_{l})=P_{2} when w_{kl}=2. Hence, substituting from (26) into (34) yields
which can be rewritten as follows
Since P_{1}(α_{31})=P_{1}(α_{24}) and P_{1}(α_{13})=P_{1}(α_{42}), the upper bound on the error probability that is equal to righthand side of (40) would be
Derivation P _{2}
In order to derive P_{2}, Eq. (37cl_given+hTacx >) is considered for b_{1}=[ 0 0]^{T}, b_{4}=[ 1 1]^{T} as follows:
Let \(\gamma _{sd} = d_{sd}^{ m}\frac {E_{s}}{N_{0}}\) and \(\gamma _{rd} = d_{rd}^{ m}\frac {E_{r}}{N_{0}}\), then
where \(x=h_{rd}^{2}\gamma _{rd} + (h_{sd0}^{2} + h_{sd1}^{2})\gamma _{sd}\). In order to find P_{2}, we average (43) over the distribution of x. Since all channels have the same distribution, {∀k,l:w_{kl}=2}, we have
The distribution of x is given by [24]
Substituting from (45) into (44) for f_{X}(x) yields
In the Appendix, we derive closed form expressions for the integrals of (46). Applying these derivation results to (46) yields
where
where \(\Gamma _{rd} = \sqrt {\gamma _{rd}/(1+\gamma _{rd})}\) and \(\Gamma _{sd} = \sqrt {\gamma _{sd}/(1+\gamma _{sd})}\)
Derivation P _{1}(α _{kl})
In order to derive P_{1}(α_{kl}), Eq. (37cl_given+hTacx >) is first considered for b_{1}=[ 0 0]^{T} and b_{3}=[ 1 0]^{T} and then we perform generalization as follows
Hence, P_{1}(α_{13}) can be derived by averaging (51) over the distribution of \(x = ({h_{sd1}}^{2} + h_{sd2}^{2}){\gamma _{sd}} \). Since all channels have the same distribution, {∀k,l:w_{kl}=1}, we have
where distribution of x is given by [24]
Substituting from (53) into (52) yields
where b_{kl}= log(α_{kl})/4. When the error probability of the SR link is less than fifty percent (i.e., P_{esr}<0.5), which is a reasonable assumption, and according to (26), we have
Accordingly, and after applying the derivation results provided in the Appendix to (54), we would have
where
and
Assignment and positioning of relay node
As will be discussed in Section 6, the derived upper bound on the error probability is tight and, accordingly, can be used to solve the relay assignment and positioning problems. In the problem of partner assignment, when a user (source) has many possible partners (relays) to choose from, the upper bound is calculated for each one and the relay node that achieves the minimum UB is selected as a partner. In the case of a fixed relay employed by the system, it is clear from (41), (47), and (56) that the error probability depends on average SNR’s γ_{sr} and γ_{rd} of SR and RD links and, accordingly, on the lengths of these links, i.e., d_{sr} and d_{rd}. When the relay node is positioned close to the destination (far from the source), the good quality of the RD link reduces the endtoend error probability; however, the poor quality of the SR link increases the error probability at the rely node and, accordingly, the endtoend error probability. When the relay node is positioned close to the source (far from the destination), the poor quality of the RD link increases the endtoend error probability; however, the good quality of the SR link reduces the error probability at the rely node and, accordingly, the endtoend error probability. Therefore, there should be an optimal position of the relay node that minimizes the endtoend error probability.
Referring to the derivations of Section 4, (41) can be rewritten as functions of source transmit SNR E_{s}/N_{0}, relay transmit SNR E_{r}/N_{0}, length SR link d_{sr}, length SD link d_{sd}, length RD link d_{rd}, and the path loss exponent m as follows:
Although the relay node can be positioned at any point in the geographical area surrounding the destination and the source, it is straightforward to show that the error probability is minimized if the relay is positioned on the straight line connecting the source and the destination and, hence, d_{sd}=d_{sr}+d_{rd}. Accordingly, (59) can be written as follows
Accordingly, parameters of the right hand side of (60) can be optimized to minimize the BER. In this paper, we are interested in studying the effect of relay node position on the error probability and in finding the optimal position that minimizes that probability when all other parameters are given. Therefore, (60) can be written as follows
Accordingly, the positioning problem turns out to be an optimization problem where d_{sr} is to be estimated for minimum P_{UB}. The problem can be modeled as a search problem in one dimensional search space and numerical solution can be used to find d_{sr} that minimizes the upper bound.
Numerical result and discussions
In this section, we present numerical results for both analysis and simulations. Although the ACMAP is a twostage decoder, the simulation results show that the decoding time of the ACMAP takes only 18.4% of the traditional MAP decoding time. In all results, we assume that the lengths of the SD link d_{sd} and the SRD link d_{rl} are equal (i.e., d_{sd}=d_{sr}+d_{rd}) and the path loss exponent is m=3.5.
Figure 3 compares the bit error probability obtained from simulations with the upper bound given by (41). The BER is plotted against the length of SR link d_{sr} (where 0≤d_{sr}≤d_{sd}) at different values of source and relay transmit SNR’s E_{s}/N_{0} and E_{r}/N_{0}, respectively. The source transmits from two antennas each with energy E_{s}/2. Three different cases are considered when E_{s}>E_{r},E_{s}<E_{r}, and E_{s}=E_{r}. We find that the derived upper bound is very tight and almost coincides with the exact error probability obtained from simulations. We also find that, in each case, there is an optimal value of d_{sr} at which the BER is minimized. The optimal value of d_{sr} depends on the values of E_{s} and E_{r}.
Figure 3 also shows other two important results. The first, under a total transmit power constraint, it is better to allocate more power to the source than the relay while positioning the relay closer to the destination. The reason is that more source power improves the reliability of both the SD and SR links while the reliability of the RD link is improved by making the relay closer to the destination. The second is related to the case when E_{s}=E_{r} which shows that the optimum relay position is not at the middle of the SRD link; however, it is closer to the destination. The reason is that the source is equipped with two antennas which makes the quality of SR link better than that of the RD link.
Figure 4 shows the minimum BER at the optimal d_{sr} against the source transmit SNR (E_{s}/N_{0}) at different values of the relay transmit SNR (E_{r}/N_{0}). The figure shows that although the SR link is not error free the ACMAP decoder is capable of achieving the expected diversity order of three. The figure also shows that increasing the relay transmit power enhances the BER performance dramatically at lower values of E_{s}/N_{0} while provides a little improvement at higher values of E_{s}/N_{0}. That is because when E_{s}/N_{0} is small, the relay is located close to the source (far from the destination) and it receives the data with high reliability, hence, retransmitting this data with high power would reduce the endtoend error probability. However, when E_{s}/N_{0} is large, the relay is located close to the destination and, hence, increasing relay transmit power would not provide that much improvement in the RD link and, accordingly, in the endtoend error probability.
In order to study the effect of E_{s} and E_{r} in the optimal position of the relay node, a numerical approach is used to find d_{sr} for different setups as explained in the following two experiments. Figure 5 shows the optimal value of d_{sr} against the relay transmit SNR E_{r}/ N_{0} at different values of the source transmit SNR E_{s}/ N_{0}. The relay node is to be positioned on the straight line connecting the source and the destination. The length of the SD link is d_{sd}=2m and the path loss exponent is m=3.5. We find that when E_{r}<<E_{s}, the optimal position of the relay d_{sr}>d_{sd}/2 (i.e., the relay has to be close to the destination node). That is because the source energy E_{s} is large enough such that the communication in the SR link is reliable even when d_{sr} is large. However, the smaller value of E_{r} requires the positioning of the relay close to the destination to increase the receive SNR of the RD link.
Figure 6 shows the optimum value of d_{sr} against the source transmit SNR E_{s}/ N_{0} at different values of the relay transmit SNR E_{r}/ N_{0}. We find that the optimal d_{sr} increase as the source transmit SNR E_{s}/N_{0} increase for a given value of E_{r}/N_{0}.
Conclusions
In this paper, we presented the MAP and the ACMAP decoding schemes for the MISO wireless relay network that adopts the DF as a cooperation protocol. The results and discussions through the paper show that the ACMAP is an optimal decoder as the MAP but with much less complexity. We derived a closed form expression for the upper bound on the bit error probability. The numerical results show that the upper bound expression is very tight. The derived closed form expression can be used for the solution of power allocation, partner assignment, and relay positioning problems. In this paper, we used the UB to study the optimal position of the relay node.
Appendix
In this Appendix, we derive the solution of integrations denoted by I_{11},I_{12},I_{21}, and I_{22}
Derivation of I _{11} and I _{12}
Let
Integrating (62) using integration by parts where u\(=\text {erfc}(\sqrt {x}+b / \sqrt {x}), dv = e^{x/\gamma }dx, du = \frac {1}{\sqrt {\pi x}}\left (1\frac {b}{x}\right) e^{(x+b)^{2}/x}dx\), and v=−γe^{−x/γ}, yields
where
and \(\Gamma \,=\, \sqrt {\gamma /(1+\gamma)}, d \,=\, b\sqrt {1+1/\gamma }\), and r = 2b(1−1/Γ). Evaluating the integral of (64) yields
where
and
Derivation of I _{21} and I _{22}
Let
Integrating (68) using integration by parts where \(u=\text {erfc}(\sqrt {x}+b / \sqrt {x})\), dv=xe^{−x/γ}dx,v=−γ(γ+x)e^{−x/γ}, and \(du = \frac {1}{\sqrt {\pi x}}\left (1\frac {b}{x}\right) e^{(x+b)^{2}/x}dx\), yields
where
Evaluating the integral of (70) yields
where
Substituting for the values of d and r into (72) and performing some mathematical manipulations yields
Substituting for the values of d and r into (72) and performing some mathematical manipulations yields
Availability of data and materials
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Abbreviations
 AC:

Alamouti combining
 AF:

Amplify and forward
 ARQ:

Automatic repeat request
 AWGN:

Additive white Gaussian noise
 BER:

Bit error rate
 BPSK:

Binary phase shift keying
 DF:

Decode and forward
 EGC:

Equal gain combining
 MAP:

Maximum a posterior
 MISO:

Multiple input single output
 ML:

Maximum likelihood
 MRC:

Maximum ratio combining
 SC:

Selection combining
 SNR:

Signaltonoise ratio
 STBC:

Space time block code
 UB:

Upper bound
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Acknowledgements
The author would like to thank Dr. Fares Almehmadi with the department of Electrical Engineering at University of Tabuk for his helpful comments and suggestions. The author, however, bears full responsibility for the Paper.
Funding
The research is supported in part by the Deanship of Scientific Research (DSR), University of Tabuk, Tabuk, Saudi Arabia, under Grant no. S14370003.
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TAK proposed the ideas and outlines for theoretical analysis, reviewed the analysis and numerical results, and revised and prepared the final manuscript. HM performed the theoretical analysis and simulations and prepared the first version of the manuscript.Both authors have read and approved the final manuscript.
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Khalaf, T.A., Mohammed, H. Bit error rate performance analysis of ACMAP in multiple input single output wireless relay network. J Wireless Com Network 2020, 12 (2020). https://doi.org/10.1186/s1363801916338
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Keywords
 MISO relay network
 MAP decoder
 BER
 Upper bound